Theorem pltfval 16959

Description: Value of the less-than relation. (Contributed by Mario Carneiro, 8-Feb-2015.)

Hypotheses

Ref	Expression
pltval.l	|- .<_ = ( le ` K )
pltval.s	|- .< = ( lt ` K )

Assertion

Ref	Expression
pltfval	|- ( K e. A -> .< = ( .<_ \ _I ) )

Proof of Theorem pltfval

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pltval.s	. 2 |- .< = ( lt ` K )
2		elex 3212	. . 3 |- ( K e. A -> K e. _V )
3		fveq2 6191	. . . . . 6 |- ( p = K -> ( le ` p ) = ( le ` K ) )
4		pltval.l	. . . . . 6 |- .<_ = ( le ` K )
5	3, 4	syl6eqr 2674	. . . . 5 |- ( p = K -> ( le ` p ) = .<_ )
6	5	difeq1d 3727	. . . 4 |- ( p = K -> ( ( le ` p ) \ _I ) = ( .<_ \ _I ) )
7		df-plt 16958	. . . 4 |- lt = ( p e. _V |-> ( ( le ` p ) \ _I ) )
8		fvex 6201	. . . . . 6 |- ( le ` K ) e. _V
9	4, 8	eqeltri 2697	. . . . 5 |- .<_ e. _V
10		difexg 4808	. . . . 5 |- ( .<_ e. _V -> ( .<_ \ _I ) e. _V )
11	9, 10	ax-mp 5	. . . 4 |- ( .<_ \ _I ) e. _V
12	6, 7, 11	fvmpt 6282	. . 3 |- ( K e. _V -> ( lt ` K ) = ( .<_ \ _I ) )
13	2, 12	syl 17	. 2 |- ( K e. A -> ( lt ` K ) = ( .<_ \ _I ) )
14	1, 13	syl5eq 2668	1 |- ( K e. A -> .< = ( .<_ \ _I ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   _I cid 5023   ` cfv 5888   lecple 15948   ltcplt 16941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-plt 16958

This theorem is referenced by:  pltval  16960  opsrtoslem2  19485  relt  19961  oppglt  29654  xrslt  29676  submarchi  29740